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in 1798, when Gauss was 21, and published in 1801, when he was 24. It had a revolutionary impact on number theory by making the field truly rigorous and systematic and paved the path for modern number theory. In this book, Gauss brought together and reconciled results in number theory obtained by
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Gauss also writes, "When confronting many difficult problems, derivations have been suppressed for the sake of brevity when readers refer to this work." ("Quod, in pluribus quaestionibus difficilibus, demonstrationibus syntheticis usus sum, analysinque per quam erutae sunt suppressi, imprimis
328:
of proper primitive binary quadratic forms, and conjectured that he had found all of them with class numbers 1, 2, and 3. This was later interpreted as the determination of imaginary quadratic number fields with even discriminant and class number 1, 2, and 3, and extended to the case of odd
270:
was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.
309:. Many of Gauss's annotations are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; they can now be read as containing the germs of the theories of
223:(Latin: 'General Investigations on Congruences'). In it Gauss discussed congruences of arbitrary degree, attacking the problem of general congruences from a standpoint closely related to that taken later by
177:. Although few of the results in these sections are original, Gauss was the first mathematician to bring this material together in a systematic way. He also realized the importance of the property of unique
258:. An English translation was not published until 1965, by Jesuit scholar Arthur A. Clarke. Clarke was the first dean at the Lincoln Center campus of Fordham College.
219:
Gauss started to write an eighth section on higher-order congruences, but did not complete it, and it was published separately after his death with the title
290:) set a standard for later texts. While recognising the primary importance of logical proof, Gauss also illustrates many theorems with numerical examples.
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in 1902 for class number one). In section VII, article 358, Gauss proved what can be interpreted as the first nontrivial case of the
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These sections are subdivided into 366 numbered items, which state a theorem with proof or otherwise develop a remark or thought.
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continued to exert influence in the 20th century. For example, in section V, article 303, Gauss summarized his calculations of
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The inquiries which this volume will investigate pertain to that part of
Mathematics which concerns itself with integers.
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333:, this more general question was eventually confirmed in 1986 (the specific question Gauss asked was confirmed by
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99:, so he did not use this term. His own title for his subject was Higher Arithmetic. In his Preface to the
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196:; Section V, which takes up over half of the book, is a comprehensive analysis of binary and ternary
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of constants. Ideas unique to that treatise are clear recognition of the importance of the
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From
Section IV onward, much of the work is original. Section IV develops a proof of
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brevitatis studio tribuendum est, cui quantum fieri poterat consulere oportebat")
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was the starting point for other 19th-century
European mathematicians, including
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Gauss, Carl
Friedrich (1889), "Allgemeine Untersuchungen ĂĽber die Congruenzen",
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197:
429:
728:
568:
Goldstein, Catherine; Schappacher, Norbert; Schwermer, Joachim (2010-02-12),
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178:
47:
410:
298:
236:
216:, i.e., can be constructed with a compass and unmarked straightedge alone.
512:
Gauss, Carl
Friedrich (1863), "Disquisitiones generales de congruentiis",
165:
Sections I to III are essentially a review of previous results, including
530:, translated by Maser, Hermann, Berlin: Julius Springer, pp. 602–629
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The
Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae
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The
Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae
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Goldstein, Catherine; Schappacher, Norbert; Schwermer, Joachim (2010),
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310:
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235:. The treatise paved the way for the theory of function fields over a
208:, which concludes by giving the criteria that determine which regular
372:(in French), translated by Poullet-Delisle, A.-C.-M., Paris: Courcier
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21:
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Gauss, Carl
Friedrich (1966) , Groth, Paul; Bressi, Todd W. (eds.),
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209:
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186:
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60:
37:
618:"Gauss' Class Number Problem For Imaginary Quadratic Fields"
545:"First Dean of Fordham College at Lincoln Center Dies at 92"
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was one of the last mathematical works written in scholarly
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Carl
Friedrich Gauss' Untersuchungen über höhere Arithmetik
380:
Carl
Friedrich Gauss' Untersuchungen über höhere Arithmetik
596:, New York, New York: Springer-Verlag, pp. 358–361,
71:, while adding profound and original results of his own.
383:(in German), translated by Maser, H., Berlin: Springer
91:. Gauss did not explicitly recognize the concept of a
447:, and His Contemporaries in the Institut de France",
657:, New York, New York: Springer-Verlag, p. 110,
405:, translated by Clarke, Arthur A., New Haven: Yale,
189:), which he restates and proves using modern tools.
103:, Gauss describes the scope of the book as follows:
152:Various Applications of the Preceding Discussions
726:
594:A Classical Introduction to Modern Number Theory
87:and parts of the area of mathematics now called
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652:
626:Bulletin of the American Mathematical Society
705:(Latin original) (first ed. 1801) (ed. 1870)
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591:
717:has original text related to this article:
701:has original text related to this article:
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29:
442:
424:; Corrected ed. 1986, New York: Springer,
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443:Dunnington, G. Waldo (1935), "Gauss, His
204:. Finally, Section VII is an analysis of
126:The book is divided into seven sections:
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221:Disquisitiones generales de congruentiis
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395:
385:; Reprinted 1965, New York: Chelsea,
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16:1798 textbook by Carl Friedrich Gauss
361:(in Latin), Leipzig: Gerh. Fleischer
200:. Section VI includes two different
341:for curves over finite fields (the
329:discriminant. Sometimes called the
13:
655:Rational Points on Elliptic Curves
14:
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183:fundamental theorem of arithmetic
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692:
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653:Silverman, J.; Tate, J. (1992),
142:Congruences of the Second Degree
721:(French translation) (ed. 1807)
640:10.1090/S0273-0979-1985-15352-2
592:Ireland, K.; Rosen, M. (1993),
506:* Latin text, with endnotes by
377:Gauss, Carl Friedrich (1889) ,
366:Gauss, Carl Friedrich (1807) ,
348:
136:Congruences of the First Degree
55:such eminent mathematicians as
25:Title page of the first edition
616:Goldfeld, Dorian (July 1985),
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543:Vergel, Gina (3 August 2009),
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355:Gauss, Carl Friedrich (1801),
303:Peter Gustav Lejeune Dirichlet
1:
493:
449:National Mathematics Magazine
274:The logical structure of the
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7:
770:19th-century books in Latin
703:Disquisitiones arithmeticae
687:Disquisitiones Arithmeticae
445:Disquisitiones Arithmeticae
399:Disquisitiones Arithmeticae
358:Disquisitiones Arithmeticae
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43:Arithmetical Investigations
32:Disquisitiones Arithmeticae
10:
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515:Carl Friedrich Gauss Werke
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430:10.1007/978-1-4939-7560-0
719:Recherches arithmétiques
522:Translated into German:
369:Recherches Arithmétiques
85:elementary number theory
74:
167:Fermat's little theorem
147:Indeterminate Equations
89:algebraic number theory
740:1801 non-fiction books
735:1798 non-fiction books
411:10.12987/9780300194258
315:complex multiplication
282:statement followed by
206:cyclotomic polynomials
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109:
95:, which is central to
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26:
194:quadratic reciprocity
173:and the existence of
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24:
760:Carl Friedrich Gauss
750:Prose texts in Latin
689:at Wikimedia Commons
331:class number problem
157:Sections of a Circle
149:of the Second Degree
52:Carl Friedrich Gauss
50:written in Latin by
243:, and a version of
185:, first studied by
155:Equations Defining
46:) is a textbook on
343:Hasse–Weil theorem
339:Riemann hypothesis
241:Frobenius morphism
139:Residues of Powers
133:Numbers in General
122:Modular arithmetic
27:
755:Mathematics books
685:Media related to
664:978-0-387-97825-3
603:978-0-387-97329-6
579:978-3-642-05802-8
486:978-3-642-05802-8
438:978-0-387-96254-2
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307:Richard Dedekind
181:(assured by the
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322:Disquisitiones
295:Disquisitiones
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81:Disquisitiones
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697: Latin
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633:(1): 23–37,
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572:, Springer,
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349:Bibliography
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299:Ernst Kummer
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237:finite field
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83:covers both
80:
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42:
41:
31:
28:
18:
311:L-functions
288:corollaries
266:Before the
729:Categories
715:Wikisource
699:Wikisource
494:References
262:Importance
233:Emil Artin
145:Forms and
120:See also:
131:Congruent
554:13 April
508:Dedekind
225:Dedekind
210:polygons
116:Contents
69:Legendre
65:Lagrange
469:3028190
280:theorem
661:
600:
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436:
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335:Landau
231:, and
229:Galois
187:Euclid
67:, and
57:Fermat
621:(PDF)
465:JSTOR
403:(PDF)
284:proof
256:Latin
93:group
75:Scope
61:Euler
38:Latin
659:ISBN
598:ISBN
574:ISBN
556:2024
481:ISBN
434:ISBN
415:ISBN
387:ISBN
320:The
313:and
305:and
293:The
250:The
212:are
79:The
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
